Towards a functorial description of quantum relative entropy

TL;DR

This paper explores a functorial approach to quantum relative entropy, extending classical Bayesian characterizations to the quantum domain, and provides preliminary calculations indicating its potential for a similar functorial description.

Contribution

It offers the first steps towards a functorial characterization of quantum relative entropy, building on classical and Borel space results, and proves it as an affine functor in finite cases.

Findings

Quantum relative entropy can be characterized as an affine functor.

Preliminary calculations suggest similar functorial properties as classical cases.

A non-commutative disintegration theorem is instrumental in the proof.

Abstract

A Bayesian functorial characterization of the classical relative entropy (KL divergence) of finite probabilities was recently obtained by Baez and Fritz. This was then generalized to standard Borel spaces by Gagn\'e and Panangaden. Here, we provide preliminary calculations suggesting that the finite-dimensional quantum (Umegaki) relative entropy might be characterized in a similar way. Namely, we explicitly prove that it defines an affine functor in the special case where the relative entropy is finite. A recent non-commutative disintegration theorem provides a key ingredient in this proof.